Abstract

We discuss contrast formation in a propagating x-ray beam. We consider the validity conditions for linear relations based on the transport-of-intensity equation (TIE) and on contrast transfer functions (CTFs). From a single diffracted image, we recover the thickness of a homogeneous object which has substantial absorption and a phase-shift of -0.37radian.

L. D. Turner, K. F. E. M. Domen,W. Rooijakkers and R. E. Scholten, School of Physics, University of Melbourne 3010, Australia are preparing a manuscript to be called ???Holographic imaging of cold atoms???

L. D. Turner, K. F. E. M. Domen,W. Rooijakkers and R. E. Scholten, School of Physics, University of Melbourne 3010, Australia are preparing a manuscript to be called ???Holographic imaging of cold atoms???

Figures (2)

Inverse of the contrast transfer functions for the TIE (blue dotted line) and CTF (red line) forms calculated for an infinite grid with spatial feature size of 1.9µm and the experimental conditions. The green dashed line is at the experimental distance. The embedded movie shows the inverse CTFs as shown here with a slider indicating the z position corresponding to diffraction distance. Also shown in the movie is a plot of an input amplitude (green) and the amplitude retrieved using either the CTF (red) or TIE (blue) methods for the indicated propagation distance. The CTF method correctly accounts for the contrast reversals that arise on propagation. The TIE method should only be applied for z closer than the first contrast reversal; it may retrieve inverted amplitudes if applied at greater z. [Media 1]

(a) CTF-retrieved thickness map for the square with grid lines.(b) Column-average of retrieved thickness for the grid pattern in the region shown in (a) for the TIE solution (blue) and the CTF solution (red). The AFM result (green) shows excellent agreement. AFM measurements also confirm the presence of grid lines outside the square. These are not a retrieval artefact, unlike the circular fringes around the contaminant at centre right. The contaminating material presumably violates the assumption of an homogeneous object.